ECON 252: Financial Markets (2008)
|Transcript||Audio||Low Bandwidth Video||High Bandwidth Video|
Financial Markets (2008)
ECON 252 (2008) - Lecture 23 - Options Markets
Chapter 1. Options Vocabulary and the 1720 Stock Market Crash [00:00:00]
Professor Robert Shiller: Today I want to talk about options. I should just say what an option is. I’ll write the word. It’s a contract that has an owner and the owner of the option contract has rights to find in the contract either to buy or sell some thing — let’s say a share of stock — at a specified price and specified date. There are two kinds; there’s a put and a call. A put option is the right to sell. It’s typically a hundred shares, so we’ll say a hundred shares of a company; let’s say it’s Google. The option would have — if it was a put option and there was a price, then you would have the right up — let me see, there’s the exercise price, also known as the strike, and there’s the exercise date.
I should also emphasize that there are two kinds of options. There are American, so called, and European, so called. It has nothing to with whether they are in America or Europe because in Europe they trade both American options and European options and in America they trade both American options and European options; so, it’s very unfortunate terminology. The American — what this means — an American option means the right to exercise the option on any date until and including the exercise date; with European, it’s only on exercise date. That’s what those words mean.
So, usually we’re talking about American options. If you have an American option — American put option — on shares of some stock, then you have the right anytime you feel like it, until the exercise date, to sell that option at the price specified in the contract, called the exercise price. If it’s European, you have to wait until the exercise date and then you have one day when you can do that. A call option is the right to buy a share of stock or whatever it is — whatever is specified in the option. In a traditional option, there are two parties; there’s the buy of the option and usually we present them from the perspective of the buyer of the option. The buyer of the option pays a price to buy the option — not to be confused with the exercise price — and then, depending on whether it’s American or European, has until the exercise date to exercise the option; but, the buyer doesn’t have to do anything. You can just do nothing; you can buy the option and if you do nothing it becomes worthless because the only way the option ever gives you value after you buy it is if you exercise it, meaning you say, I will use my right to buy or sell.
The other party is the writer of the option. Because it’s a contract, it has to be between two parties. Somebody is on the other side and you can do either one; you can either buy or write an option. If you — let me make this clear; if you write a call option, then what you are committing yourself to do as the writer — you sign the contract from the writer’s contract, which goes along with the buyer’s contract. Well, it provides rights to the buyer. If you write a call option, then you — and say it’s American — then you are signing a contract — let’s say it’s on stock — to deliver one hundred shares to the other guy, the buyer, whenever that guy feels like it. That guy will pay you the contracted price, so it’s not — it doesn’t seem like much fun to be a writer of an option because you have — you’re just sitting there waiting for this other person to make up his or her mind. There’s a benefit; mainly, you get the money. The buyer of the option pays you up front for providing this right to the buyer, so writers of options write them hoping that they expire unexercised; that’s when they make money. If you write an option and the buyer of the option pays you the money up front and then you never hear from the buyer again, then you’re — that’s the way you like it. So, you make money by writing options and hoping that they don’t get exercised.
Of course, you can write a put option and that means — if you write a put option, you are signing a contract that says that whenever this other guy on the other side, the buyer, decides to, that guy will sell you a hundred shares at the specified price. Again, you’re laying yourself open to, whenever this guy wants to, you’ve got to receive a hundred shares and pay the money.
Now, these kinds of contracts are very old and, in fact, we had a conference over the weekend at the Yale School of Management on — it was a very interesting — I’ve never experienced anything quite like it. Maybe I should put the website up for you to look at. There’s a book; it’s called The Great Mirror of Folly, written in 1720 about the stock market and the Beinecke Rare Book Library has a copy of it. They’re very rare — about the stock market crash of 1720. Did you know that there was a big stock market crash in the year 1720? What was happening in New Haven in 1720? Well, I know one thing that was happening in 1720 in New Haven — I’m guessing; I’m pretty sure. You had some pretty angry investors who lost everything in the stock market, but it couldn’t have been the U.S. stock market, which wasn’t created yet. The crash of 1720 was primarily in Paris and London, also less so in Amsterdam; those were the financial centers of the world. So, I’m speculating there must have been someone here in New Haven; probably Yale University lost in this crash — I don’t know. There must have been someone here who lost — it was a huge and devastating stock market. This is the first one actually; the first stock market crash.
We had a lot of fun at this conference; it just relates to options. I’ll tell you why it relates to options, because people were writing options galore in 1720 on stocks. The book — if you search on Great Mirror of Folly on the Web, it’ll come up with our conference and proceedings. Since this book — copyrights expire after, well, it’s a complicated formula, but in less then a century. So, this is all public domain, so Yale has it up on the Web and you can read the whole book. Unfortunately, it’s written in Dutch, which might deter some of you, but it has lots of pictures. We had great — at this conference, the — it was the most interdisciplinary conference I’ve ever seen because we had professors from the Art History, Comparative Literature, Finance, Economics, and Psychology. We had scholars from all over the world who knew about the year 1720, including a lot of Dutchmen who were here.
Anyway, the highlight of it was — one highlight for me was, it had a — we saw a picture of an option from this time — an option contract to buy stocks from Amsterdam. It showed it was a printed form; they had printed forms back then, at least in Holland they did. So, a printer had printed up with blanks to fill in. There’s a place to fill in the exercise price and the exercise date. I don’t know whether it was American or European at the time, but I’m sure if it was American they didn’t call it an American option in 1720. They didn’t even call them options. Of course, it’s all in Dutch, so I don’t know; it was some other word — not options. I’m just saying this because the —
The other interesting thing about 1720 is that they didn’t make the same distinction between investing and gambling that we do now. Right now, anyone on Wall Street is very loathe to have any suggestion of connection with gambling, but back then they didn’t care. So, lots of stocks would have lotteries attached or there would be all kinds of — something called a tontine, where a group of investors would invest in something and then all the money would go to the last one to die, after all of them died but one. That’s a sort of gambling; I don’t know what sense it makes, but they did that.
I remember an old story on — I just heard this somewhere — from the 1920s. Two brokers on the New York Stock Exchange floor were talking to each other and one of them says, “I’ll be you $5 that the market’s going to go up.” Then the senior man scolded him and said, “are you betting? You will be thrown off this floor permanently if I hear that word again.” So, that attitude has persisted — that investing should be distinguished from gambling. I suppose there’s good reason for that because gambling instincts can take hold of us and investing has a good purpose. Unfortunately, our emotions can carry us away from the good purpose and gambling is not investing. Back in 1720, the distinction was not so clear and this event was so — it got — the reason they called the book The Great Mirror of Folly is that the event got totally crazy. People were squandering their life fortunes.
I’ll tell you one more story. We had a great time at this conference because this book, Mirror of Folly, includes plays that were written in 1720 and performed in Amsterdam about the crash — about the stock market crash. The organizers of this conference got some students from Saybrook College to perform one of the plays from Great Mirror of Folly. Is anyone here from Saybrook? Okay, you weren’t in the — I didn’t see you there though. There was a scene in the play where the young woman was being told by her father that he intends for her to marry a very promising young man who is speculating in stocks and will soon be rich. She is very skeptical about being forced to marry; she has somebody else in mind. The father says that the other young man is worthless; he’ll never amount to anything. But, she stuck by her guns and insisted that I will never marry a man who’s in love with the stock market. We don’t know what happened because it’s all fictional, but as we know, the whole stock market crashed, so she was right on two counts probably.
Chapter 2. The Standardization and Logic of Options: Options Exchanges [00:14:58]
Anyway, that’s all about — options are very old, but they’ve emerged more recently as very important contracts. In particular, what they didn’t have in 1720 — in fact, they didn’t have anywhere until recently — is an options exchange. The problem with a traditional option is that it’s a contract between two parties. If you write — if you buy an option, you’re at the mercy of this other person. So, if you buy an option that was written by a broker, as they were in 1720, what if the other guy doesn’t — he just skips town; he’s gone. You bought this option to either buy or sell and then when the date comes you can’t find this guy. So, what do you do? You obviously were cheated out of your money. So, we created — that was a problem until 1973, when the first options exchange opened. Well, I think there may have been ways of dealing with the problem, but not before ‘73. But, this is the first options exchange — Chicago Board Options Exchange — which was a spin off of the Chicago Board of Trade.
Now, what they did was they organized a central marketplace for standardized options. Options used to be written for whatever exercise date anybody wanted; there was no standardization. An options exchange is like creating a futures market when you only had a forward market in the past. They started trading options on U.S. stocks in 1973 and they require that the writer of a naked call has to put up margin. What is a naked call? If you write a call, you are standing ready to sell a hundred shares to the buyer, whenever that buyer decides — if it’s American — to do that. But you’re naked if you don’t own the hundred shares. One way you can do it is, you can show that you own a hundred shares, so there’s no way that you could fail to deliver. If you’re naked, then you are required to put up margin and the margin is an amount that was enough so that if you fail to deliver, the CBOE could access your margin account and buy the shares on the market to sell to the buyer; there’d be enough money to do that. The margin requirement for the writer makes the contract secure so that there is really no counterparty risk with options purchased on an options exchange. Now, there are many options exchanges, but the CBOE — I’m just listing it — was the first.
Now, futures exchanges sell options on futures; that’s the same thing as an option on a stock, but instead of a stock contract, it’s a futures contract. That would be done at the CME Group, which is a futures exchange. That’s just — we’re just talking about where you can do these things. Did I explain the concept of options? Maybe I should go through the — I have here a plot illustrating a call option. On the vertical — on the horizontal axis, I have stock price; that’s $0 a share, $5 a share, $10 a share. I’m showing it up to $45 a share. Now, I’m going to illustrate the intrinsic value of an option with a $20 strike price. Now, the option would be typically for 100 shares, but I’m going to describe it as if it were an option to buy one share; so, it would be 1/100th of a typical option. This broken straight line is what we call the intrinsic value of the option, which is the money you could get if you exercised it right now. If you decided — we’ll never have intrinsic value negative because you wouldn’t exercise.
So, let me explain what this means. Suppose you own an option with an exercise price of $20 and the price of a share is $15. What is the value of that option today — the intrinsic value? Well, it’s nothing because the option gives me a right to buy a share at $20, but hey, I can buy it under the stock market for $15, so I would never exercise the option today. It would be worthless; it would be worth minus $5 if I exercised it today because I would be paying $20 for something I could get for $15, but I’m not going to call it minus 5; I’m going to call it 0 because you just — you won’t exercise. If the stock price is below the exercised price, I have a value. On the vertical axis is the intrinsic of the call; I have to distinguish it between actual value. This isn’t the price that is quoted for the option; this is what it would be worth if you exercised it today if it were — the option has value beyond its intrinsic value because even though it’s worthless today, it might be worth something in the future. I’ll come back to that but this is just — I’m just talking about intrinsic value.
Now, what if the stock price is $30 today. What is the value of the option — the intrinsic value? Well, it’s going to be $10, obviously, because if you exercise it today, you’re buying the stock for $20 and you can sell it today for 30 on the stock market; so, the difference is 10. This line here — it doesn’t look like a — this is a 45º angle here — doesn’t look like it, but that’s what it is. It has a slope of one. It’s very simple; the intrinsic value for a call is just a broken straight line; it breaks at the exercised price. To give you a little bit more jargon, in this region we say the option is “out of the money.” That means, exercised today, it would be worthless. This one right here is called “at the money.” If the stock price is equal to the exercised price, then the stock — we’d call the option at the money, for a call. Here, if the stock price is above the exercise price, we say it’s “in the money.” This line goes off into infinity; I just stopped it there. That’s straightforward, right?
Anyway, this does illustrate something about options that is different from anything we’ve discussed before. This is a broken straight line, not a straight line. All of our talk about portfolios to date has been linear. When you combine stocks, you are making your portfolio respond linearly to the return of any one of the stocks in the portfolio, but this is non-linear because we have a break; that’s what options do, so it’s non-linear finance.
Some people are confused about what options really are. Often people say, well if I buy a stock option that means I can make up my mind later whether I want to buy and sell. So hey, I’m just getting the right to be indecisive or to — well, think of it this way — I haven’t made up my mind whether I really want to invest in options or not — in stocks or not — so, I’ll buy an option and that gives me the right to buy. You could say that and a lot of people think that way. Like, a company will think, we’re trying to decide whether we want to build this shopping center. So, we’ll buy an option on the land underlying where we would build the shopping center and we’ll think more about it and decide whether it’s a good idea to build a shopping center. You could do that, but there’s something a little bit misleading about that reasoning because whether or not you decide to build the shopping center, if you buy an option on the land you will always exercise it if it’s in the money on the exercise date — whether you build a shopping center or not.
Suppose you couldn’t decide whether to build a shopping center and you bought an option on land and then someone comes in and says, well we have to make up our mind today; the option is exercising — is expiring — if we don’t exercise it today it’s worthless. What do you discuss at your meeting? You don’t discuss whether we’re going to build the shopping center or not; that’s irrelevant. You discuss, what can we sell the land for and if we can sell it for more than the exercise price, we will always exercise it. So, there’s no — the assumption in finance is that all options that are in the money on the exercise date are exercised and there’s no choice. The word option might be misleading because — you could choose to be dumb and not exercise it, but that’s not what it’s about.
On the other hand, options really are central to our thinking about a lot of things. I’ll give you an example of an option that you might not consider an option. This is the option to marry somebody. Sometimes people will complain that their boyfriend or girlfriend cannot commit. We’ve been going out for three years; it’s time that we get married, but this person — the counterparty — cannot seem to decide. Actually, one view of it — of that situation — could be that this person is just better schooled in finance than the other because one principle of finance is that you should never exercise an American call early. I’m not this cynical about relationships; I’m just telling you a story that comes to mind. You’d never want to — I’ll come back to that — you never want to exercise an American call early, so that’s why there isn’t an important distinction between European and American. Just in the case of relationships, suppose your girlfriend or boyfriend really wants to marry you and is still giving you time, then you instinctively know you should wait as long — I’m saying not really, but I’m saying in terms of theory; you should wait until the last day when this other person says it’s now or never because there’s always exercise — there’s always option value. There’s always a chance — maybe that will become clearer. I don’t know if you like my analogy. I’m not cynical about these things as some people are. Peter?
Chapter 3. The Put-Call Parity Relation [00:27:57]
Professor Robert Shiller: This is a call option up here, but — this is a call. I’m going to show you a put — go ahead. What arbitrage —
Professor Robert Shiller: The price — I’m going to come back to that. The price of the option will always be above that line, so there’s no arbi — there are possible — it depends on if the price is wrong — they’re arbitraged. Let me come back to that. This is a put option. This is intrinsic value for a put option because it’s the opposite of a call. If the strike price is $20 and the stock price is selling for $15, then you can see that it’s in the money because you can make $5 by exercising. You have the right to sell it for 20 but you can buy it in the market for 15, so you buy it for 15 and sell it for 20; you make $5. On the other hand, up here — if the stock price is $30, you have the right to sell it for $20. Well, that’s worth nothing; I can sell it for 30 in the market.
Then what — let me jump to this; this isn’t exactly the order that I wanted to do it, but the — this has to do with arbitrage as you were saying. The price in the market should always be greater than the intrinsic value, until the exercise date — the last date for an American — even American or European, but let’s talk American. This pink line is my price for the option; we’ll talk about how we get that line from theory, but price — if the — let’s say, if it’s an American option it’s got time to go. Let’s say the exercise date is not for another year and it’s out of the money; the price of a share is only $15 but the exercise price is $20. That option is still worth something today, right? It’s not worthless. It would be worthless if you exercised it today, but hey, you’re not going to exercise it today. The reason it has value is that the price might rise above $20 sometime over the next year and so it — you have a chance of making money on it. That means the price of the option is always going to be worth more than the price of the stock. No, it’s always going to be worth more than the intrinsic value.
What about at the money? An at-the-money option — if the price of a share is 20 and the option — and the exercise is 20, exercise it today and it’s worthless. It has to be valuable because fifty-fifty chance the stock price is going to go up and so you have a good chance of making money on it. So, it’s going to be worth a lot more than this one was down here because we’re at the money. Any little jostle upward is going to put it in the money. There’s a big chance that this will become in the money, so it has real value; whereas, down here, the option doesn’t have much value because it would take a big price move to put it in the money. Then what about up here? Even when they’re in the money they’re worth more than intrinsic value. You kind of wonder, well why would that be? Well, it’s because this thing is better than owning the stock. Let’s say, at this point, when the stock price is 25, I’d rather own the option than to own a share minus $20 because the option can’t fail me as much as the share can. The share minus $20 could be negative in value before the exercise date, but the worst that can happen to my option is it would be worth nothing.
The arbitrage, Peter, that you were referring to is the arbitrage — what if the price were below this line? Maybe that’s what you were thinking. What if the option price — what if the stock price is $30 and the option is selling at $5? If it’s an American option I have an immediate arbitrage. I buy — if it’s a call option I would buy the option for $5. I would exercise it and sell for $30 and I’ll make money instantly. That can’t happen. We can’t have the option price selling for less than intrinsic value, so you know just from arbitrage that that pink line is always above the solid line. Moreover, there’s another arbitrage relation which I didn’t show on the chart, but if we draw a 45º line here from the origin, that’s plotting the stock price against the stock price; it has a slope of one and it comes out of the origin. No option can be worth — can be priced up above that 45º line — above here. In other words, an option can never sell for more than the stock. Does that sound obvious? Why would you pay — if the share is selling for $25, why would I pay $30 for the right to buy at that $20? Obviously it’s ridiculous; the stock itself is an option to buy the share at a $0 exercise price, so it has to be worse to have a positive exercise price.
Now, I wanted to stress the put-call parity relation and this is another arbitrage thing. The put option price — arbitrage — the absence of arbitrage opportunities implies that the put option price minus the call option price equals the present value of the strike price — that’s discounting it from this exercise date to the present — plus the present value of any dividends coming between today and the exercise date, minus the price of the stock. This has to hold because if it didn’t hold, there would be an arbitrage opportunity. The put — let me just show you why. This diagram is supposed to explain that. I’ve got here the intrinsic value of the call, which is the yellow line, and this is an intrinsic value of a put. We’ve got them both at the same exercise price. Then, I’ve shown here the stock price on the blue line; stock price against stock price is just a 45º line — a line with a slope of one. Well, you notice that if I were to buy a call and write a put — that’s the same thing as shorting a put — I would have a combined portfolio with just those two. I would have the yellow line here and I’d have minus the pink line here. I would have a parallel straight line that looks just like the stock price just shifted down. If I buy a call and short a put — or write a put — it’s the same thing as owning the stock minus the exercise price.
So, that’s what we have in the put-call parity relation, so that we’re taking account of dividends. That diagram didn’t show the fact that stocks will pay — might pay dividends between now and the exercise date. You can see what I was just saying. I’ve got — I said call minus put. Well, this is put minus call, minus the price of stock. I have a minus sign in front of everything here, but it’s just what that diagram shows. What put-call parity means is that we only need a theory of either call prices or put prices and then the other one falls right out of put-call parity. All I need is a theory of call prices, so we will just forget about puts and we’ll just talk about calls from now on.
Chapter 4. Pricing an Option: The Black-Scholes Formula [00:36:32]
If I give you a problem to ask you — what is the put price — what is the price of a put? You would go in and calculate the price of a call and then calculate the put option price. We’ll put this on right hand of the equation. The put price would equal the call price plus the present value of the strike price, plus the present value of dividends, minus the price of the stock; so, that makes it very easy. All we have to do is worry about calls. Where am I? The question for financial theory is, what determines this pink line? You agree that it should be above intrinsic value. As the option gets closer to expiration — as time moves on and the exercise date is getting closer and closer in time, this pink line is going to go down, down, down. On the last day, it hits the intrinsic value.
What is it before the exercise date? I’m going to start with a theory, which illustrates how we calculate these things. This is a theory that applies to — so that we can understand it easily — that applies to a strip-down situation. I’m going to derive the price of an option under the assumption that it’s very simple. There’s only one period between now and exercise; it’s a European option. We’re going to exercise it in one — we have an exercise date of one period and also under the restrictive assumption — and this is for pedagogical purposes just to simplify option theory — that the stock price, S, is the stock price today. The stock — this stock is very special because next period it can have only two values. It’s S times u if the stock goes up; u stands for up. It’s S times d if the stock price goes down. What I’m saying — what’s arbitrary here is, I’m saying that there are only two possible prices for the stock next period — Su or Sd. That’s not real world because as you know there are all kinds of infinite number of possible prices next period. Again, this is just — I think that we should be able to figure out what the price of a call option on this stock should be worth. It’s very simple. They say it’s very simple, but the people who invented this won the Nobel Prize for this, so I won’t — I don’t want to make it — this wasn’t so simple in the history of financial thinking.
Do you understand the situation that we’re proposing in? It’s just like — there’s this very funny stock that we know — for some reason we know that S is the price today and next period, when the option exercise date is, its price is either going to be Su or it’s going to be — S times u — or it’s going to be S times d. Then, there’s an interest rate and we can both borrow and lend at this riskless interest rate. What should the option be worth? In this case, I’m going to call C the current price of the call today. Now, this is before the exercise date, so the price of the call is going to be worth more than the intrinsic value. I’m going to call Cu the value of the call next period if the price is up and Cd the value of the call next period if the price is down.
That’s the thing that we read off of those broken straight lines. So, Cu would be the stock price minus — it’s the stock minus the exercise price if it’s in the money. We know that in advance because we already know what the two possible prices are next period; so, we already know what the two possible option values are next period. This is the intrinsic value if it’s up and this is the intrinsic value if it’s down. We’ll call E the strike price of the exercise price of the option. Is everything clear here? It’s just such a very simple world. I’m just saying there are only two possibilities; it’s a very simple world and there’s only period between now and exercise, so it’s very simple.
Now, what I’m going to say — we’re going to develop an arbitrage theory of options and we’re going to say that you want to — you’ll take any profit opportunity that’s riskless. It ought to be possible to get a riskless profit opportunity here by investing both in the stock and the option because there are only two possible values for the stock and you’ve got both a stock and an option. There must be a riskless portfolio because the price of the option depends only on the price of the stock on the exercise date. What I’m going to do is get an optimal hedge ratio, H, that makes my portfolio. I’m going to form a portfolio of the stock and the option and I’m going to put them together so that I have a riskless portfolio; that’s what I’m going to do. Out of that is going to fall a value for the price of the option. This is what you want to do.
We’re looking for a riskless profit opportunity. Let’s consider this, we’re going to write one call and buy H shares and I’m going to pick H so that I have no risk at all. It’s easy to see how you do that because we already know — before the exercise date — we know that if the price goes up it will be worth uHS — my portfolio — if I have H shares, the H shares will be worth uHS. uS is the price; H shares will be worth uHS. I’ve written one call so this would be worth uHS minus the price of a call. Similarly, if the price stock — if the stock price goes down, then this is the value — intrinsic value of the call, then next period on the exercise date the portfolio will be worth dHS minus CD. Let’s choose H so that the two are the same. All I have to do is set this equal to this and solve for H and that gives me the optimal hedge ratio. So, H is equal to Cu – Cd, all over (u – d) x S. Now it’s very simple to get to option pricing.
If I can form this portfolio where I have [short] one call and [long] H shares in a portfolio, it’s a riskless portfolio, so it has to earn the riskless rate of interest. That’s what no arbitrage assures. It can’t be possible to get a riskless portfolio that earns either more or less than the riskless rate because if that did happen I would have a riskless opportunity — ability to earn more than the riskless rate with no risk. That’s contrary to arbitrage. The return on — since you invested HS - C in the portfolio, the return on it — the total value of it — has to equal (one plus the riskless rate) times (HS – C). If you substitute in for HS – C, you find out that it equals this. Substitute for H into this and you get the price of the call today. It’s simple algebra, but there it is.
So, that’s the arbitrage theory call option price. That might be less than intuitive to you. See, that was very simple arguing that got us there. We merely said, the way to think about options is that options move with the stock price and they’re perfectly correlated with the stock price over this interval because if the stock price goes up you know you’ve got Cu. If the stock price goes down, you know you’ve got Cd. So, you have only one source of uncertainty but you have two assets, so you can put them together to eliminate risk. If you put them together that way they have to earn the riskless rate and you just solve for it and you get this value for the call option. This is the inherent insight that Black and Scholes came up with in their classic 1973 paper on option pricing, which I’ve come to, but this has to be the price of the call option in this simple world; otherwise, there would be arbitrage.
The interesting thing about this is that there are no probabilities in this formula. What’s in this formula? I’ve got the riskless rate; I’ve got intrinsic value; I’ve got the difference between the price and the two circumstances, but nothing to do with probabilities. This puzzled people. People thought, well doesn’t the price of an option have to depend on the probability that it will come in the money? If it’s an out of the money option, don’t I weigh the probability? It’s not in this formula. You might say, it’s implicitly in the formula because the relationship of S to Su and Sd involves probabilities, but it’s not in this formula. Black and Scholes, in their famous paper, used this kind of reasoning to get to the standard option contract, which — option formula. I’m not going to derive it because the mathematics is quite a bit more difficult, but it’s exactly the same logic that I just went through with the binomial option pricing formula. This is one of the most famous formulas in all of finance. What Black-Scholes did is, under certain assumptions about the stochastic properties of stock prices and under the assumption of no arbitrage opportunity, they came up with a formula that an option price should follow if there’s no arbitrage.
I’m just going to present their formula and then we’ll think more about options. The Black-Scholes formula — call T the time to exercise. Before, I just said it was — when we talked binomial, I just had — I said it was one period hence, but now we’re allowing the exercise date to be any distance in the future. So, T is the time to the exercise date. This is for our European option, although it’s often used to apply to American options as well. We’ll call σ2 the variance of the one-period price change. N(x) is the cumulative normal distribution function, which you can find on Excel; it’s called normdist, so I don’t want to get into these details. This is the formula that Black-Scholes won the Nobel Prize for. Well actually, Black died at a relatively early age from throat cancer; he was a heavy smoker and people don’t do that anymore, so that’s one risk that is over. Scholes won the Nobel Prize for this — they don’t award that posthumously — for this little formula.
It says, the price of a call is equal to the stock price times N(d1) minus the exercise price times N(d2), where d1 is given by this expression and d2 is given by this expression. That might not be intuitive to you. We could spend a couple of lectures making that more intuitive, but I’m just going to stop with that formula now. That’s the formula that I used back here to make this pink line. I just — I had to plug in a value for σ2 and T, but I did that and I used the Black-Scholes formula. The Black-Scholes formula does what we sort of think it should. The price of an option should be greater than the intrinsic value everywhere, but here is the exact equation.
This is one of the most famous equations in finance. It might even be on your — if you have a financial calculator you might have a key that you can press. It’s already on your laptop and you don’t even know it probably — maybe — depending on what kind of programs you have. It’s easy to compute in Excel; you just have to use this normdist. This cumulative normal distribution function is not something you can do by hand; you have to use a — it would involve an integral that doesn’t have an analytic solution, but you can get it on Excel.
Chapter 5. Accounting for Volatility in the Black-Scholes Formula [00:51:35]
Now, the critical problem with the Black-Scholes formula, however, is getting some of these parameters that have to go into it and the tough one is σ2; most of the other things we know. If you’re trying to actually price an option using Black-Scholes, S, we already know that’s just the stock price. E is the exercise price; we know that — that’s written in the option contract. T is the time to expiration date; we know that — that’s written in the contract. r is the riskless interest rate; well, that’s easy to tell — that’s just quoted in the market. There’s only one thing that remains that’s tough and that’s σ2. That’s the variance of the stock price.
Black-Scholes says you have to know how variable the stock price is to price an option and intuitively you can see it. Isn’t it obvious that the more variable the stock price the more valuable the option is? If the variance were zero, then the option would just be the intrinsic value because there’s no chance for the stock to do anything unexpected. If it’s out of the money and the variance is zero, the option is worthless. If it’s in the money and the variance is zero, then it’s worth something, but it’s only worth the intrinsic value. If σ2 is 0, the price can’t move anywhere, so there’s no problem. As σ2 increases, the option gets more and more valuable; if it’s out of the money it’s getting more and more chance to come into the money and so that’s in the formula.
The key variable in the Black-Scholes formula is the variance of option — of the underlying stock price; that’s the kicker; that’s the hard part. People who trade options use the Black-Scholes formula, but there’s a problem and the problem is you’ve got to plug in a number for σ2. So, what number should I plug in? Well, you might say, let’s take historical numbers. I know pretty much what the variance of stock price changes is; let’s use the historical variance. I wanted to show you the historical variance of stock prices. I have — since I like history, I go all the way back to 1871. What I did to compute this chart is I took the S&P composite or S&P 500 Index back to 1871. This is my spreadsheet, which is on the web under our classroom materials. I took a six-month moving average of six-month changes — six-month standard deviation of the percentage price change for every month from 1871, July, to April of 2008.
The important thing to understand here is that the variance is not constant through time; it moves around. There are high variance periods for stock prices and low variance periods. I like to look at this picture because it’s interesting. The first thing that is interesting is that, overall, the market has been remarkably consistent for over a hundred years. The variance back in the nineteenth century doesn’t look any different; nothing has changed in a hundred and thirty years; the market has always been volatile. There’s only one thing that jumps out at you when you look at this picture and that’s these numbers in here. You notice those numbers in the middle? That was the Great Depression of the 1930s and something went really haywire in the financial markets in the Depression. This is 1929; it wasn’t so much 1929, but remember this is a six-month — it’s a lagging six months. So, where does it — maybe the 1929 crash is here somewhere. I’m not sure exactly where it is, but something went really haywire after 1929 and the markets got extraordinarily volatile for a while. That was a crisis period in American history that shows up really well on this picture, but since then nothing much has happened.
It’s interesting to look recently. This doesn’t look very important; it shows how important our lifetimes in the broad sweep of history, but your lifetimes are from in this region right here. The thing that’s interesting is that we have been recently in a very low volatility period for the stock market. This is around 2003. We were in a high volatility period in the 1990s — nothing like in the Great Depression, but high in the ’90s and then volatility just collapsed and the markets were the deadest and dullest place to be in the world — the stock market. Partly, I think this is because our — we were — who knows why this was. I’m going to throw out a wild suggestion. It’s because we were distracted by the housing bubble and all talk after the stock market peaked in 2000. Lots of people just lost interest in the market and all of their speculative enthusiasm was focused on housing and the stock market was kind of forgotten. There has to be at least an element of truth to that story, but something has been happening lately. Look how much volatility is shooting up now; this is because of the world financial crisis that we’re in. If you look at this picture it doesn’t look like the world financial crisis is very important compared to historical events so far that we’ve seen in the past.
Now, I want to define what we call implied volatility. What the — you can do with the Black-Scholes formula is it requires that you input σ2 to calculate what the price of an option should be. Why don’t we take what the price of the option is and work backwards to figure out what volatility is implied by the price. Do you see what I’m saying? You can solve — you can turn the Black-Scholes — the Black-Scholes formula gives the price of the option in terms of σ2. Well, I can turn it around because I know what the market price of the option is; they’re traded on the CBOE. So, I would take the market price of an option and turn it around and get what the implied volatility is from Black-Scholes. The CBOE does this for you because they trade S&P 500 options and they have it on their website and its called VIX, so I have VIX there. That’s their volatility index.
The CBOE was created in 1973. Unfortunately, the series doesn’t — it goes only back to 1986, but it’s been going for a long time. You can’t get implied volatility back to 1871 because — although there were options traded back then, there was no organized market. You can’t get a consistent series of prices of options going that far back, so you can’t get implied volatility. Maybe you could do it if you got some records from some broker and find some option, but it would be hard. Our implied volatility only goes back a little over twenty years, but the interesting thing is I have plotted here both the implied volatility over that period and the actual volatility. You see that they line up fairly well, but this shows the strength of the Black-Scholes formula. Black-Scholes does seem to be pricing options well enough because the implied volatility, while it’s not perfectly exactly equal to the actual volatility, it’s close and so we can see that the formula makes some sense.
Chapter 6. Options on Home Prices as Risk Management [01:00:08]
I just wanted to conclude this lecture by talking about the effort we’ve made to get single-family home price options going at the Chicago Mercantile Exchange. As I told you, I and some of my colleagues have been campaigning with futures exchanges to create futures markets for home prices and, ultimately, for commercial real estate prices and other economic variables. We went to — we started campaigning almost twenty years ago, but recently we have been talking with the Chicago Mercantile Exchange and they created futures markets for single-family homes in May of 2006 using the S&P Case-Shiller Indexes. That was our objective, but when we went there the CME said, well why don’t we start options as well — options on futures. So, we have a futures contract and they launched options on home prices. You can see all these things on the Chicago Mercantile Exchange website. What we have now are both puts and calls for ten U.S. cities. Unfortunately, they’re not doing well; they’re not selling well, but they’re still going and we’re hoping that we can get them to connect. Often markets are slow at first, so I hope that that’s the story — that it’s just a slow beginning.
Let me say what we have; maybe it’s a good down to earth way to illustrate the value of options. I was telling you how to price options, but I didn’t tell why would you buy one. Well, I did tell a story at the beginning. I talked about an investor wanting to invest money in options and a writer hoping that the option will expire unexercised and hoping to make money. I didn’t really go behind that and emphasize why you would do that. I think the example of a home price option is very easy to see.
Let’s consider a situation that someone — I don’t think any of you are homeowners here at this point; maybe some of you are — probably not. But imagine — try to imagine that you bought a house a couple years ago and now the housing market is collapsing. You bought that house probably on a mortgage, so you borrowed 80 to 90% or maybe even more of the money to buy the house. Now, the home is falling and you’re thinking, hey wait a minute. This house is worth less than my debt and you start to get upset. You’re thinking, well you know I’d like to move to another city, but then you realize, I can’t do it. If I sell my house, I won’t be able to pay off the mortgage; I’ll be bankrupt before I — I can’t move. This is very real. In fact, economy.com, a consulting firm, estimates that in the United States today 10% of all homes are underwater in that sense. The mortgage debt is greater than the value of the home and that number is increasing everyday as home prices continue to fall.
Some people are very upset. So, what can they do? Well, one thing they can do or they could have done a couple of years ago if they had thought to do it, they could buy a put on the home — on homes in the city where they have a house. I put in, say, around — let’s say I put in $500,000 into a house and I’m worried that its price might fall. Well, if you buy a house and you buy a put on that house together, the two together eliminate your risk.
Let’s think of it as just buying a put on a house rather than on an index. I buy the house for $500,000; I put up $400,000 and so — I put up $100,000; the mortgage puts up $400,000. I’m underwater if the house drops $100,000. I’ve lost — my mortgage debt exceeds my — I don’t want that to happen, so why don’t I just buy a put on the house, which is a right to sell the house at $400,000 until some exercise date; then, I can’t possibly be underwater. If I ever decide to move or if it’s an American option, I can just — if the price of my house is less than $400,000, I’ll just exercise my put. There’s no way for me to get wiped out. In fact, I could buy a put at $450,000 and that way I would always be sure that I’d have $50,000 left. So, that’s the idea of using options as a hedging mechanism.
While I gave an example in terms of real estate and it’s not widely used that way, this same idea is used a lot by investors in other domains, so I think options have a very real risk management purpose. In a sense, an insurance contract is like a put option. If I buy fire insurance on my house, then it’s like buying a put option on my house, but it’s only exercisable if there’s a fire. What it says is, if my house burns down, the put option — I can sell whatever remains at a price, which is determined by the insurance contract; it’s the same as a put option. Insurance is not fundamentally different from finance. We’ve had a little trouble deciding whether we want home equity insurance or just puts — home equity puts — so we’ve created the puts. At some point, we want to also create insurance on homes, someday; I hope that happens. There is no home equity insurance.
These are just different incarnations of the same risk management ideas. The fundamental idea here in finance is that you can create — options are examples of derivatives; I should add that a derivative is a financial contract that derives from another financial. So, an option is a derivative because the price of the option in the options market depends on the price of something else in another market — the stock market. So, our real estate options are another example of derivative. The price of the put option depends on — in the option market — depends on the price of the house in the housing market. One of the themes in my forthcoming book, which I’m writing right now, is that derivatives are like insurance. They’re fundamentally important to risk management vehicles and they could have helped prevent the subprime crisis that we’re now in if they had just gotten established and more developed.
[end of transcript]Back to Top
|mp3||mov [100MB]||mov [500MB]|